Question
Find $x$, if $: 4^{2 x}=\frac{1}{32}$
✓
Answer
$4^{2 x}=\frac{1}{32}$
$ \Rightarrow(2 \times 2)^{2 x}=\frac{1}{2 \times 2 \times 2 \times 2 \times 2}$
$ \Rightarrow\left(2^2\right)^{2 x}=\frac{1}{2^5}$
$ \Rightarrow 2^{2 \times 2 x}=2^{-5}$
$ \Rightarrow 2^{4 x}=2^{-5}$
We know that if bases are equal, the powers are equal
$\Rightarrow 4 x=-5$
$ \Rightarrow x=\frac{-5}{4}$
$ \Rightarrow(2 \times 2)^{2 x}=\frac{1}{2 \times 2 \times 2 \times 2 \times 2}$
$ \Rightarrow\left(2^2\right)^{2 x}=\frac{1}{2^5}$
$ \Rightarrow 2^{2 \times 2 x}=2^{-5}$
$ \Rightarrow 2^{4 x}=2^{-5}$
We know that if bases are equal, the powers are equal
$\Rightarrow 4 x=-5$
$ \Rightarrow x=\frac{-5}{4}$
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